Choosing the closest weather station:

The Canadian government’s past weather and climate service offers a search by proximity function. Via this service, some sample data of the closest stations to Montreal were downloaded. Each of the data files contains information about the weather stations, including the geographical position (latitude and longitude). These coordinates were plotted on a map (see Figure on the right), and the closest station to the bulk of the data was chosen (station name: MCTAVISH). Only data from this station was used.

Figure: Plot shows all starting stations of a bike trip (red dots), as well as the closest weather stations (blue markers). The closest station in the center is the MCTAVISH weather station (Climate Identifier 7024745, WMO Identifier 71612)

To get a better understanding of the data, the number of hourly bike trips was visualized for the time span between \(2014\) and \(2017\).

The moving average that is shown in the plot (red line) can be interpreted as a baseline model, i.e., the simplest possible model to describe the hourly number of bike rides.

This baseline model explains \(38.8\%\) of the variance \((r^2 = 0.388)\) and has a mean absolute error of \(MAE = 316.2\), which means that on average, the “prediction” for the number of hourly bike rides is wrong by this many bike rides. This includes also winter months with no rides. For a more realistic estimation of model quality, these numbers drop to \(r^2 = .079\) and \(MAE = 510.7\) when only considering the time frame from May to September.

Figure: Number of hourly rides from \(2014\) to \(2017\). Each dot represents the number of trips in one specifc hour. Red line represents a moving average using a window of \(14\) days.

To visualize the relations between the available features, a correlation heatmap is shown on the right. The features are only slightly correlated, with the only exception being temperature and dew point that show an almost perfect (linear) relationship \((r = .93)\).

To avoid problems resulting from this multicollinearity, only temperature was used as a predictor, and dew point was dropped. Despite the fact that that gradient boosting is less influenced by multicollinearity, it might still influence calculations of variable importance [2,3].

Figure: Pearson Correlations between available features in the data.

Table: Model performance measures for different models. \(r^2\) is the amount of variance explained, \(MAE\) stands for mean absolute error. \(train\) and \(test\) specify training and testing set, respectively. For the test set, some performance measures were also re-computed for using only the summer months of the test set (May to September), indicated via the \(summer\) subscripts.

Model	\(r^2_{train}\)	\(r^2_{test}\)	\(r^2_{summer}\)	\(MAE_{test}\)	\(MAE_{summer}\)
Gradient Boosting (XGBoost)	\(0.889\)	\(0.860\)	NA	\(158.0\)	NA
Random Forest	\(0.913\)	\(0.894\)	NA	\(111.2\)	NA
Gradient Boosting (sklearn)	\(0.997\)	\(0.941\)	\(0.933\)	\(85.4\)	\(105.4\)

The explained variance of the different models tried ranged from \(r_{test}^2 = 0.860\) to \(r_{test}^2 = 0.941\) for the final model (all in the test set; see table). The hyperparameters for the final model (gradient boosting via scikit-learn) were selected via randomized search using \(4\)-fold cross validation (using only the training set).

The final model fits the data very well, explaining \(94.1\%\) of the variance and exhibiting an average error of \(85.4\) rides over all hourly predictions. This error only increases to \(105.4\) if only summer months are considered.

Figure: The figure shows actual (true) values vs. predicted values for the number of hourly bike trips for the final model. A perfect model would yield predictions that are identical to the true values, i.e., all points would be on the \(45°\) diagonal. This model is relatively close.

The most important features that influenced the prediciton of hourly bike rides were temperature and atmospheric pressure. While the former is easily comprehensible, the latter is best understood as a proxy for precipitation, which was not available from the weather data: low pressure is commonly related to rainy weather [4]. While Relative humidity is not as tightly connected to rain [5], it interacts with temperature, e.g., it influences how (high) temperatures are perceived [6].

Further important predictors are the hour of the day and the day of the week, as well as wind direction and speed. How these features influence the predicted number of bike trips is best detailed by specific plots, so-called Partial Dependence Plots (PDP) and Individual Conditional Expectation (ICE) plots [1].

Figure:

Individual Conditional Expectation (ICE) plots [1] to quantify the main effects of some of the predictors.

For details, see figure caption.

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Individual Conditional Expectation (ICE) plots [1] to quantify the main effects of some of the predictors.

For details, see figure caption.